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Population -
entire group of interest
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Example of a 'population'
Ex: All cats in Wake county, All mortgages owned by a bank, All people in the US
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Parameter-
measure associated with the population (usually pose a research question about a parameter)
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Example of a 'parameter'
Ex: (True) average weight of all cats in Wake county,
(True) Proportion of mortgages that will default,
(True) standard deviation in income for all people in the US
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Sample -
subgroup of the population data is collected on
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Statistic -
measure associated with the sample
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Example of a 'statistic'
Ex: (Sample) average weight of 40 cats from Wake county, (Sample) proportion of mortgages that defaulted last year, (Sample) standard deviation in income for 1000 random selected people in the US
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Set -
- A collection of ‘elements’, x1; x2; ...
- Notation:
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Sample Space -
Ω (capital Omega, also denoted as S) - the set of all possible outcomes of an ‘experiment.’
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Example of a 'sample space'
- Examples:
- - Toss a coin
- - Record pet types
- - # of pets a person has
- - Lifetime of a light bulb in hours
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Union of A and B: (A ⋃ B)
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Intersection of A and B: (A ⋂ B)
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When are sets A and B disjoint?
- When they are mutually exclusive
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Partition -
A 1, A 2,... A n form a partition of Ω if
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- and the Ai’s are pairwise disjoint
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Difference in sample average for experimental and control groups
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How do you determine if the average difference (D^) is significant?
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What is the Central Limit Theorem (CLT)?
often the average of a large number of random variables has a bell-curve distribution
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Some facts about sets:
- - order is not important in a set
- - repeated elements can be removed
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What does the following notation mean?
x1 ∈ A
x1 is in set A
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What are the two ways that a set is notated?
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What does Ω denote?
a set
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Event-
any expected outcome from a Ω
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What is a countable set?
A set in which all elements can be listed
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What is an uncountable set?
You are unable to list all the elements, they are infinite
ex. integral
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What happens when you have a pairwise disjoint set?
If you look at the intersection of the sets, the resulting set is null/empty
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Examples of distributive Laws of sets:
A⋃(B⋂C)=
A⋂(B⋃C)=
A⋃(B⋂C)=(A⋃B)⋂(A⋃C)
A⋂(B⋃C)=(A⋂B)⋃(A⋂C)
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Example of DeMorgan's Law for sets:
(A⋃B)c=
(A⋂B)c=
(A⋃B)c=Ac∩Bc
(A⋂B)c=Ac⋃Bc
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Random Variables (RV)-
varies from sample to sample, but is numeric
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Distribution-
pattern and frequency with which we observe our values
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What is the definition of Probability that we will be focusing on?
Relative frequency of an event in a repeated experiment
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What is a probability measure on Ω?
- It is a function P from events of Ω to the real numbers that
- satisfies:
- Axiom 1: P(Ω) = 1
- Axiom 2: If A ⊂ Ω then P(A) ≥ 0
- Axiom 3: If A1, A2, · · · are (mutually) pairwise disjoint then
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What is Axiom 3 of a probability measure on Ω?
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What is Axiom 2 of a probability measure on Ω?
Axiom 2: If A ⊂ Ω then P(A) ≥ 0
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What is Axiom 1 of a probability measure on Ω?
Axiom 1: P(Ω) = 1
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How does Axiom 3 give a great way to define probabilities when we have a countable sample space (two steps)?
- 1. Define all ‘sample points’ (elements, call these Ei)
- 2. Assign appropriate probabilities to each sample point
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What are the properties of the Consequences of Axioms?
Property A: Complementary Law, P(Ac) = 1 − P(A)
Property B: P(∅) = 0
Property C: If A ⊂ B then P(A) ≤ P(B)
Property D: Addition Law, P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
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What is the multiplication principle?
If a job has k tasks, each done in n1, n2, ... nn ways, respectively the job can be done in total n1, n2, ... nn ways
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Permutation-
an ordered arrangement of distinct objects
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Factorial-
- for a positive interger (n or 0), we define:
- n!= n*(n-1)*(n-2)...*3*2*1
- 0!= 1
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Permutation Rule
- The # of permutations of n distinct
- objects taken r at a time
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Binomial coefficient rule
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Overview of counting ideas, and how to choose r objects from n distinct objects
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Counting idea when:
Order matters
without replacement
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Counting idea when-
Order matters
with replacement
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Counting idea when-
Order doesn't matter
without replacement
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Counting idea when-
Order doesn't matter
with replacement
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multi-nomial coefficient rule
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What is conditional probability?
updating one's beliefs based on obtaining new knowledge
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